A Nash equilibrium of a game is a set of (possibly mixed) strategies $\sigma=(\sigma_1,\ldots,\sigma_n)$ such that, if each player $i$ believes that that every other player $j$ will play $\sigma_j$ then $i$ should play $\sigma_i$ That is, when $u_i$ is the utility function for the $i$ th player:

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Translated, this says that if any player plays any strategy other than the one in the Nash equilibrium then that player would do worse than playing the Nash equilibrium.